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Properties

Label 2-338-13.12-c3-0-28
Degree 22
Conductor 338338
Sign −0.969+0.246i-0.969 + 0.246i
Analytic cond. 19.942619.9426
Root an. cond. 4.465714.46571
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 8.74·3-s − 4·4-s − 14.1i·5-s − 17.4i·6-s − 28.6i·7-s − 8i·8-s + 49.4·9-s + 28.3·10-s − 9.49i·11-s + 34.9·12-s + 57.3·14-s + 123. i·15-s + 16·16-s − 30.6·17-s + 98.8i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.68·3-s − 0.5·4-s − 1.26i·5-s − 1.18i·6-s − 1.54i·7-s − 0.353i·8-s + 1.82·9-s + 0.896·10-s − 0.260i·11-s + 0.841·12-s + 1.09·14-s + 2.13i·15-s + 0.250·16-s − 0.436·17-s + 1.29i·18-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=((−0.969+0.246i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+3/2) L(s)=((−0.969+0.246i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: −0.969+0.246i-0.969 + 0.246i
Analytic conductor: 19.942619.9426
Root analytic conductor: 4.465714.46571
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ338(337,⋅)\chi_{338} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 338, ( :3/2), −0.969+0.246i)(2,\ 338,\ (\ :3/2),\ -0.969 + 0.246i)

Particular Values

L(2)L(2) ≈\approx 0.54568512440.5456851244
L(12)L(\frac12) ≈\approx 0.54568512440.5456851244
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1−2iT 1 - 2iT
13 1 1
good3 1+8.74T+27T2 1 + 8.74T + 27T^{2}
5 1+14.1iT−125T2 1 + 14.1iT - 125T^{2}
7 1+28.6iT−343T2 1 + 28.6iT - 343T^{2}
11 1+9.49iT−1.33e3T2 1 + 9.49iT - 1.33e3T^{2}
17 1+30.6T+4.91e3T2 1 + 30.6T + 4.91e3T^{2}
19 1+153.iT−6.85e3T2 1 + 153. iT - 6.85e3T^{2}
23 1−36.0T+1.21e4T2 1 - 36.0T + 1.21e4T^{2}
29 1+49.2T+2.43e4T2 1 + 49.2T + 2.43e4T^{2}
31 1+166.iT−2.97e4T2 1 + 166. iT - 2.97e4T^{2}
37 1+23.8iT−5.06e4T2 1 + 23.8iT - 5.06e4T^{2}
41 1+125.iT−6.89e4T2 1 + 125. iT - 6.89e4T^{2}
43 1−434.T+7.95e4T2 1 - 434.T + 7.95e4T^{2}
47 1+186.iT−1.03e5T2 1 + 186. iT - 1.03e5T^{2}
53 1+400.T+1.48e5T2 1 + 400.T + 1.48e5T^{2}
59 1+408.iT−2.05e5T2 1 + 408. iT - 2.05e5T^{2}
61 1+603.T+2.26e5T2 1 + 603.T + 2.26e5T^{2}
67 1−287.iT−3.00e5T2 1 - 287. iT - 3.00e5T^{2}
71 1−961.iT−3.57e5T2 1 - 961. iT - 3.57e5T^{2}
73 1−963.iT−3.89e5T2 1 - 963. iT - 3.89e5T^{2}
79 1+1.04e3T+4.93e5T2 1 + 1.04e3T + 4.93e5T^{2}
83 1+313.iT−5.71e5T2 1 + 313. iT - 5.71e5T^{2}
89 1−675.iT−7.04e5T2 1 - 675. iT - 7.04e5T^{2}
97 1−272.iT−9.12e5T2 1 - 272. iT - 9.12e5T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.89316099632382696977923570539, −9.799612390223754790493087026215, −8.807943823404423536844026533019, −7.45777265442682667908710421938, −6.78070033283171784965039754040, −5.69566850136974958294628499577, −4.75934981103368262033062071730, −4.23020060065756259639287806827, −0.947216223449310020092715482101, −0.32760209980606574047374358591, 1.75828585826602940925284455950, 3.12273278575924337326988128205, 4.67535049577467621875267928653, 5.82494172787040893585116761241, 6.28632692633012346121557681730, 7.58308146405157944733447923447, 9.066819091042602007458034366574, 10.15184568859025939554749098314, 10.78523234423924336653178581514, 11.49077109809930448591793382012

Graph of the ZZ-function along the critical line